In this main part, we combine the insights gained in previous sections.
In Sec. REF , we identify a necessary and sufficient criterion for bipartite entanglement (Thm. REF ).
In Sec. REF  we employ the structure theory of Jordan and (associative) \(C^*\) -algebras to translate this criterion into a compatibility condition between canonical time orientations on local subsystems (Thm. REF ). Finally, in Sec. REF  we interpret our results in light of the intrinsic flow of time in von Neumann algebras by means of Tomita-Takesaki theory, Connes cocyles, and the background-independent thermodynamical arrow of time [1]}.
